Renormalisation of Chiral Gauge Theories with Non-Anticommuting $\gamma_5$ at the Multi-Loop Level

TL;DR

This thesis develops a systematic, algorithmic approach for the multi-loop renormalisation of chiral gauge theories using a non-anticommuting $oldsymbol{ ext{γ}_5}$ scheme, achieving the highest-order calculations to date and applying it to the Standard Model.

Contribution

It introduces a fully automated, high-precision framework for multi-loop renormalisation of chiral gauge theories with non-anticommuting $ ext{γ}_5$, including the complete 4-loop Abelian case and 1-loop Standard Model.

Findings

Successful 4-loop renormalisation of Abelian chiral gauge theory.

Complete 1-loop renormalisation of the Standard Model in the BMHV scheme.

Identification of efficient prescriptions for dimensional regularisation implementations.

Abstract

This thesis presents a comprehensive study of the renormalisation of chiral gauge theories in dimensional regularisation (DReg) at the multi-loop level. We employ the mathematically consistent Breitenlohner-Maison/`t~Hooft-Veltman (BMHV) scheme with non-anticommuting $\gamma_5$, whose modified algebraic relations induce a spurious violation of gauge and BRST invariance. A central focus is the systematic restoration of the broken symmetry, for which we provide a transparent and fully algorithmic procedure based on the quantum action principle. A major achievement of this work is the complete 4-loop renormalisation of an Abelian chiral gauge theory -- the highest-order application of the BMHV scheme to date. This calculation is made possible by an automated, high-performance computational framework incorporating several optimised algorithms. Our results demonstrate that a rigorous, self-consistent treatment of $\gamma_5$ is feasible even at very high loop orders. We further analyse dimensional ambiguities and evanescent details corresponding to different implementations of the regularisation, and identify practically efficient prescriptions for $D$-dimensional fermions and gauge interactions. Building on these insights, we present the complete 1-loop renormalisation of the full Standard Model (SM) in the BMHV scheme, providing a first step towards a fully self-consistent multi-loop renormalisation of the SM and establishing a solid foundation for future high-precision electroweak phenomenology.